1547671870-The_Ricci_Flow__Chow

28 2. SPECIAL AND LIMIT SOLUTIONS

Hence by (2.8), g must have the form

1

g = ds^2 + 2 tanh^2 (as) d(J2

a

for some a > 0. We claim that g is nothing other than a constant multiple

of the cigar soliton. To se13 this, make the substitution <Y = as to get

g = 1 (^2 2 2)^1

(^2) a d<Y + tanh <Yde =^2 a 9L.·
REMARK 2.8. By (2.13), we have f' = ar.p and hence
1

f ( s) = - log ( cosh (as)).

In particular

a

a

x = - grad f = -a<p. e1 = -tanh (as). as.

D

EXERCISE 2.9. Find the analog of the cigar metric with curvature K < 0.

3. Ancient solutions

An ancient solution of the Ricci fl.ow is one which exists on a maximal

time interval - oo < t < w, where w < oo.

3.1. The round sphere. The shrinking round sphere is in a sense the

canonical ancient solution of the Ricci fl.ow. Let 9can denote the standard

round metric on sn of radius 1, and consider the 1-parameter family of

conformally equivalent metrics

g (t) ~ r (t)^2 9can,

where r (t) is to be determined. Observe that g (t) is a solution of the Ricci

fl.ow if and only if

dr a

2r dt · 9can = atg = - 2 Re [g] = - 2 Re [9can] = -2 (n - 1) 9can,

hence if and only if r (t) is a solution of the ODF.

dr ---n - 1

dt r

Evidently, setting

(2.14) r (t) = Jr8 - 2 (n - 1) t = J2 (n - 1) · vT -t

yields an ancient solution (Sn, g (t)) of the Ricci fl.ow that exists for the time

interval -oo < t < T, where T < oo is the singularity time defined by

2

T = ro

· 2 (n - 1)"